Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

W ^ALTER L ^ITTMAN

Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 5, n

^o

3 (1978), p. 567-580

<http://www.numdam.org/item?id=ASNSP_1978_4_5_3_567_0>

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Boundary Theory Hyperbolic

and Parabolic Partial Differential Equations

with Constant Coefficients (*).

WALTER LITTMAN

(**)

~

dedicated to Hans

Lewy

1. -

Introduction.

Suppose

^{we have a}

well-posed initial-boundary

^value

problem

^{for a}

hyperbolic

^or

parabolic equation

^{Lu = 0 in a}

cylindrical

^{domain D = D X}

X[0, oo),

where D is a bounded domain in Rn. One basic

problem

^of

boundary

^control

theory

is that of

null-controllability: given

initial data in D at t ⁼

0,

^{can this data be}

supplemented

^with

appropriate

^non-homo-

geneous time

dependent boundary data, prescribed

^{on the lateral}

boundary

of

S~,

such that the solution of the

resulting

^initial

boundary

^value

problem

will vanish for t &#x3E; T?

Furthermore,

^{how small can T} be chosen? For the

wave

equation,

^{the heat}

equation

^{and more}

general

second order

equations

this

problem

has been treated

by Russell [9], Fattorini [2], Seidman [10]

and others. For additional references see these works.

The

present

^{work was to a}

large

^extent

inspired by

two sources, which

we are

happy

^to

acknowledge:

^A number of very instructive conversations with L. Markus which

acquainted

^us with the basic

problems

^{and work}

done in the area;

secondly,

^{a lecture}

given by

Frank Jones in November

1973,

the

subject

^{of which we} shall describe

shortly.

Once instance where the

problem

^of

boundary

^control

theory

^{has a rather}

straight-forward

solution is the wave

equation

^{in an} odd number of space

(*) Partially supported by

NSF Grant MCS 76-05853 the

University

^{of Min-}

nesota.

(**) University

^of Minnesota, ^{School of} Mathematics,

Minneapolis,

^Minnesota.

Pervenuto alla Redazione il 28

Luglio

^1977.

dimensions

greater

^{than one.}

(See

^Russell

[9]).

^One

simply

extends the initial data outside D with

support

^{in a}

slightly larger set,

^and

proceeds

to solve the pure initial value

problem

with this data. Since the Riemann function for this

problem

^has

support

^on the surface of a

(forward)

^cone

(Huygen’s principle),

the solution restricted to ,~ will vanish for t &#x3E;

T,

where T can be determined from the

geometry

^{of D. The}

appropriate boundary

data for the control

problem

^{is then}

simply

^{read off from the}

solution determined as described above. Let us note that this

procedure

^works

whenever the Riemann function has an

appropriate

^lacuna.

Frank Jones’ result

[7],

^{alluded to}

earlier,

consists in

constructing

^a

fundamental solution to the heat

equation

^{which can}

play

^{the same role}

in the

boundary

^control

problem

for the heat

equation

^{that the}

ordinary

Riemann function

plays

^{for the}

(odd

space

dimensional)

^wave

equation-a

role for which the

ordinary

fundamental solution of the heat

equation

^is

not suited. Jones constructs a fundamental solution with

support

^{in 0}

with s &#x3E; 0

arbitrary.

The methods

just

described have several

advantages.

One is that the

boundary

^control

problem

is solved once and for all without reference to any

specific

^initial

boundary

^value

problem.

Another is that one need

only

have

uniqueness

for the mixed

problem.

The existence of a solution to the control

problem

^{is a} consequence of the

procedure. (Uniqueness

of the solu- tion of the control

problem

^{is not asserted here and}

is,

ⁱⁿ

general,

^{not valid}

under our

hypotheses.)

^{A third}

advantage

^{of the}

procedure

is that it is not restricted to

cylindrical

^domains.

We

briefly

describe the content in the remainder of the paper. Section 2 reviews some

elementary

facts about

hyperbolic operators

^{for n = 1. In}

section 3 we treat the

boundary

^control

problem

^for

strictly hyperbolic equations

^{in one} space variable

(which curiously enough,

^{is not}

quite

^as

immediate as the case n

odd &#x3E; 1).

^{We dwell at}

greater length

^{on the one}

dimensional case than

might

^seem necessary

mainly

because the basis of the method is encountered here

already.

In section four we use a

plane

wave

decomposition

^{to treat} the n dimensional

hyperbolic

^{case. Section}

five deals with a’

parabolic equation

^{in one}

dimension,

^{while section}

six

deals with the n-dimensional _case,

again

^{via a}

plane

^wave

decomposition.

The main theorem for the

parabolic

^{case is stated at the end of that}

section.

This paper

emphasizes

^{methods and}

qualitative

^results.

Thus,

^{for the}

sake of

simplifying proofs,

^{we assume strict}

hyperbolicity, although

^results

continue to hold for

general hyperbolic operators

with constant coefficients with

slight

modifications in the

proofs.

^{We are} also content with

stating

all results in the context of C°°

functions,

^at least in the

hyperbolic

^case.

We intend to discuss estimates in Sobolev norms

(for

^the

hyperbolic case),

more

general equations,

variable coefficients and other

aspects

^{of the}

problem

at another time.

The

subject

of this paper ^was

presented

^at the Conference ^on Partial Differential

Equations

^{held at}

Oberwolfach,

March 1975.

2. - Some

elementary

^facts

concerning hyperbolic equations

^{in one} space dimension.

. We first treat the case of a

strictly hyperbolic operator corresponding

to a

homogeneous polynomial (i. e.,

^{with no} lower order

terms)

where we assume

that a 0 0,

^{the a’s are real}

and,

for the sake of

simplicity,

distinct. The differential

operator t

⁼

a(a/ax) -f- alat represents

^a directional derivative in the direction

(oc, 1)

^{and has two} fundamental

solutions,

g- and g+,

i.e.,

solutions to the

equation £g = (x) b(t),

^both

being

^measures

with

support

^{on the line at =} x; g+

vanishing

^for

negative t,

g- for

posi-

tive t.

Both

represent signals propagating

^with

velocity

^{ce and}

beginning

^or

ending

^{at x =}

0,

^{t = 0.}

Let us assume that none of the _cxi

vanish, i.e.,

^{there is no zero}

velocity

of

propagation.

^{Then we}

gain complete symmetry

between the x and t variables. Moreover each

operator ~i

⁼

-~- a/at

^{will have two fun-}

damental solutions each of which can be

uniquely

determined

by specifying support

either in the upper ^or lower

(closed)

^half

plane--or alternately,

in the

right

^{or left half}

plane. Convoluting

^the fundamental solutions

for Li

with

support

^{in the}

(upper, lower, right, left)

^half

plane

^we

obtain, apart

from a constant

factor,

^a fundamental solution

(G+, G_, GR, G,)

^{to the}

operator

^{L with}

support

in that half

plane.

The actual

support

^of

G+

^is

the sector in the upper half

plane

^bounded

by

the two lines

(passing through

the

origin) corresponding

^{to the}

algebraically

lowest and

highest velocity

of

propagation; min ai

^and max aii.

Letting

^{we see that}

GR(GL)

will have

support

^{in the sector in the}

right (left)

^half

plane

^bounded

by

the rays

corresponding

^{to the}

algebraically

^{lowest and}

highest reciprocal velocity: i.e., min Ai

^{and max}

Âi.

It will be of

importance

^{to us to notice}

that if we

let X

⁼

max

^{then the}

support

^of

GR(GL)

^{is contained in the}

intersection of the set

X

with the

right (left)

^half

plane.

Next we consider

operators

with lower order terms

where

Q(~, 7:)

^{is a}

polynomial

^with

complex

coefficients of

degree

^m.

L has four fundamental

solutions,

y which we

again

^denote

by Gt, GR, GL, having

^{the same}

supports

^{as those for}

P m(D).

Furthermore these will vary

continuously

with the coefficients of the

operator,

^as

long

^{as the}

speeds

^of

propagation

^remain

separated.

3. - The

hyperbolic

case, n =1.

Let L ⁼

P(Dx, Dt)

^{be a}

strictly hyperbolic operator (n = 1 ),

^{and let}

5 be the

reciprocal

of the slowest

speed

^of

propagation,

⁼

1/min

THEOREM 1. Let C°°

Cauchy

^{data be}

prescribed

^{at t = 0} in the closed in- terval

1

on the x axis. Then there exist 000 solution ^v to Lv ⁼ 0 in

1

X

[0, 00), assuming

^this

Cauchy data, vanishing for

^t &#x3E;

T 1,

^where

T 1

is any number

exceeding To =-

^diameter

1

We recall that

by C°°(I) (for

^a closed interval

Ï)

^{we mean functions}

extendable to C°° in a

larger

^interval.

PROOF. Extend the initial data to a

Co

function with

support

^{in a}

slightly larger

^interval

1,

contained in ^an s

neighborhood

^of

I,

^{and solve}

the

equation

^{Luc = 0} with the extended

Cauchy

data for all ^- oo x ~ oo, Call the solution u. Let

cp(t)

^{be a} C°° function

equal

^{to one for}

and zero for

t ~ 2 .~’1-~- E,

^{and set}

L(ugg)

If m denotes the

mid-point

^of

I,

^{let W be a C°° function}

equal

^{to one}

for and zero for

x&#x3E;m+8,

^{and set}

and

« *)}

denoting involution, and G, and GR

the fundamental solutions described in the last section. For 8

sufficiently

^small

(which

^{we now so}

choose) ~IL

and

U R

^are C°° and each vanish in

neighborhoods

of the sets

(This

^{is of course a} consequence of the estimates of the

supports

^of

GL

^and

G,

of the

previous section.)

^{Hence the sum TI = does}

also,

^{and more-}

over satisfies

Lu = f. Next,

^{set We have}

Zv = 0,

v agrees with u ^{near t =}

0,

^{has the}

originally assigned Cauchy

^{data in}

I,

and vanishes for Hence it is a desired solution.

4. - The n dimensional

hyperbolic

^case.

We shall

accomplish

the transition from one to several space dimen- sions

by

^a standard device-a

plane

^wave

decomposition.

^{To that}

affect,

let us recall some basic facts

concerning

^such

decompositions,

^{as described}

for

example by Ludwig [8].

^For

/e8(.B") (S

^{is the} Schwartz space of

rapidly decreasing

^C°°

functions)

the Radon transform is defined

by

where c~ is a unit vector and

represents 11, -

¹ dimensional surface area,

oo.

For g = g(s, w)

^define

.H

denoting

^the

(one dimensional)

Hilbert transform. Then a

plane

^wave

decomposition

is

accomplished by setting

If

f belongs

^{to then is}

infinitely

differentiable in s and c~. In

our

applications

^the

^function

^will

depend

^{on another}

parameter, t,

^and

part

of the last statement has a more

quantitative

^version:

Here v and y are

integers depending

^on the dimension

n; k

^{is an}

arbitrary

non

negative integer;

the maximum on the left is taken over all s E .R1 while the one on the

right

^{is taken over} all x E Rn. Both maxima may also be taken ^over the same closed t interval. « Const. &#x3E;&#x3E;

depends only

^{on n.}

We ^{are now}

ready

^to treat the case of a

strictly hyperbolic operator

with constant coefficients in n-space dimensions. Let L ⁼

be such ^an

operator,

^{and D a} bounded domain in Rn. For each

(spacial)

direction w

(i.e.,

unit vector in

x-space)

^let

.

^{denote the}

reciprocal

^{of the}

slowest

speed

^of

propagation

^{in the (o} direction. Let

dm

be the diameter of the

w-projection

^of

D, i.e.,

^D

projected

^{on a line}

parallel

^{to the a) vector.}

Let

To =

sup

~,~, ~ do.

m

THEOREM 2. Given

assigned

⁰⁰⁰

Cauchy

^{data at t = 0 on}

^D (which

^{can be}

extended as 000

f unctions

^{in a}

neighborhood of D) ;

^then

for

^each

Tx &#x3E; T,

^there

exists a solutions ^{v to} Lv ⁼ 0 in

D X ~t

&#x3E;

0} assuming

^the

prescribed Cauchy

data in

D

and

vanishing in (Note:

^{D =} closure of

D.)

PROOF. Let us extend the

Cauchy

^{data so as to be C°° and have}

support

in ^{an 8}

neighborhood

^of

D,

^{and let us solve the}

resulting Cauchy problem

ⁱⁿ

all of

jR"x(00), calling

^{the solution u. Let}

PI

^{be as} in the statement of the theorem and

let,

^as

before,

where is a C°° function such that

For each unit vector co let

i.e.,

^{the ro}

projection

^of

D,

^{and let m. denote the mid}

point

^of

^1w.

^Let

W(s)

be such that

and set.

be the

plane

^wave

decomposition of f,

^{and let}

We note that

where

Dt)

^{is a}

strictly hyperbolic operator

ⁱⁿ s, t space

depending

in a C°° way of (o,

having

the strict

hyperbolicity

maintained

uniformly

with

respect

^{to co. As a matter of}

fact,

^the

polynomial Pw(a, 7:)

^is

nothing

else but the

polynomial P(~, 7:)

restricted to the

2-plane in ~

^{- r} space which contains the t-axis and the ^OJ

vector,

^{with a =}

~.

From the case n ⁼ 1 we know that the

operator Dt)

⁼

Lro

has fundamental solutions

with

supports

contained in the sets

(respectively),

^where

5m

is the maximal

reciprocal propagation speed

^for

Zo.

Letting

(with

convolutions carried out in s ^- t

space)

^we notice that these func- tions are Coo

in s, t,

^{w and}

satisfy

hence

(and

^{the same with « R}

»),

^{with the}

supports

^of

U WL

^and

U WR

both con- 37 - Annali della Scuola Sup. di Pisa

tained in a

neighborhood S’

^{of the set}

Moreover

~5~, approaches ^Sm

^{as E - 0. A} moment’s reflection shows that for e

sufficiently

^small

(which

^{we now so}

choose),

^{the set}

S,

^{is bounded}

away from the sets

and hence from the sets

1-1 ’I.

and

which are contained in the above sets

(respectively).

^{It follows that the}

supports

^of

UroL

^and

UWR

^{both are} bounded away from

(*)

^and

(~~),

^uni-

formly

ⁱⁿ

Hence,

y if we define

U will be ^a C°’ solution to

vanishing

ⁱⁿ

neighborhood

^of

(*)

^and

(**). Defining (as

^{in the case n =}

1)

we see that v satisfies all

requirements

of the theorem.

5. - The

parabolic

^{case n =1.}

To motivate our

procedure,

^we rederive Frank Jones’ result for the

one dimensional heat

equation along

^{somewhat different} lines. We wish to imitate the

procedure

^{followed in the}

hyperbolic

^{case. We}

begin by

solving

the traditional initial value

problem, prescribing

^as initial data

d(x),

thus

arriving

^at the standard fundamental solution

G+(x, t),

^with

support

in t &#x3E; 0. Our next

step

^{would be to}

multiply G+ by

^a cut-off function

T(t),

this time

choosing (p(t) equal

^{to one}

for It

^{E and}

equal

^{to zero}

for It

&#x3E; 28.

To

proceed

^{with the}

analogy,

^{we then must} look for the

analogue

^of

G,

and

GR

^with

supports

ⁱⁿ

neighborhoods

^{of the}

negative

^and

positive x

^axis

respectively.

Such fundamental solutions do not exist as

distributions,

^but

they

^{do exist}

formally (see Ehrenpreis [l], p. 209)

where

6"1

is the i-th derivative

of 6(t)

^and _x+ ^{= max}

(x, 0).

This « funda- mental solution » has

support

ⁱⁿ

(x &#x3E;- 0, t

⁼

0)

and may be

interpreted

^{as a}

functional in a space of functions

satisfying

^a

Gevrey

condition of index 2 in t. A second « fundamental solution »

having support

^on

(x ~ 0, t

⁼

0),

may be obtained

by replacing

x+

by

^{x_ - Let us hasten to add}

that the existence of these

generalized

fundamental solutions is

nothing

but ^a reformulation of the observation of

Holmgren [5]

^to the effect that the

Cauchy problem

for the heat

equation

^{in one} space dimension ^can be solved if

Cauchy

^{data in an}

appropriate Gevrey

^{class is}

prescribed

^{on the}

taxis.

Digression

^on the spaces

y-5.

^From the above discussion it becomes clear that we should choose the cut-off function

99(t)

^{in an}

appropriate Gevrey

class. Instead we choose to work with a variation of

the ya

^classes.

(For

these spaces and their

properties

^{see Hormander}

[6],

^p.

146).

^{We say}

99(t)

^E

E

c 1~ belongs to y’o

^{if for every 8} &#x3E; 0 there is ^a constant

C~

such that

We

define y2 = ^t)

^as the space of continuous functions

t) belonging

for each x to

with

bounds uniform

in x, and yy 6 = ’5(x, ^t)

the space of continuous functions

belonging

for each x to

yf(t)

with bounds uniform in x on bounded subsets of x-space. We shall

always pick

¹

^{c ~}

^2.

We now come back to the

parabolic equation

where A is a one dimensional differential

operator

^{with constant} coefficients of order m &#x3E; 1.

(The procedure

^{with some}

modifications,

^y also works with

variables coefficients

depending

^{on x}

only).

^{We assume that the}

highest

order ^term in A is of even order and has

positive

coefficients. Under these conditions L has a fundamental solution

G+

^with

^support

^{in 00 which is}

analytic

for t &#x3E; 0. We now

proceed

with the construction of formal fun- damental solutions

analogous

^{to the ones} exhibited for the heat

equation. ^Denoting by T,

^{we set}

(with G,

^defined

similarly),

^{where is the}

unique

^solu-

tion to

A i+1 a --- ~ (x)

^with

support

^on 0153&#x3E;O

(0153:O).

^A

simple computation

verifies that these are indeed

formal

fundamental solutions.

As ^a convenience we shall

from

^{now on}

always

^{choose 8 so that 0 8}

l.

Having

chosen the cut-off with ⁼ 0

for t J ~ 2E

^and

= 1 for we choose E 000

equal

^{to one for x c -1 and zero for}

set

and notice that these functions

belong

^to

y~, (together

with all their x deri-

vatives)

and have their

supports

in the sets

(respectively)

We would like to define

provided

^{we can attribute a}

meaning

^{to these}

expressions,

^{and then}

proceed

as in the

hyperbolic

^case.

Formally,

x+l

For y

^{in a} bounded interval

I,

^we have the estimate

(similarly

^{for a, and} y

0).

^For

fR,L

^{we have} the estimate

with a similar estimate

holding

for any finite number of x derivatives of

f replacing f.

An

application

^of

Stirling’s

^{formula now} insures the absolute and uniform convergence of the series

defining UR L

in any finite interval I. Further- more, y the series may be differentiated with

respect

^{to x or t} any number of times without

affecting

the convergence, thus

justifying

^{the hitherto}

formal relations

Setting v

⁼

U R - U L

^{we see that Lv = 0 for}

t &#x3E; 0, v

agrees with

G+

for t ⁸ and vanishes for

t ~ 2E,

^{hence is our} desired fundamental

solution.

Let us note that an alternative to the use of

G, and GR

would have been to

apply

Theorem 5.73 of

H6rmander[6].

^However then the construction would not

generalize

^to variable coefficients

(for n

⁼

1).

6. - The

parabolic

^case n &#x3E; 1.

If we assume that .L is

parabolic

^{in the sense that}

being

^the

principal part

of A. Under these condi.

tions there exists a fundamental solution

G(x, t)

^with

support

ⁱⁿ

t &#x3E; 0,

^ana-

lytic

^{in x and t for t} &#x3E; 0. We shall use the

following

Estimates

for

^{the standard}

f undamental

^{solution :}

where

h(x) ---&#x3E; 0

faster than any

negative

^{power of}

l0153l

^as

Ixl

^{- oo, and where}

C and K may

depend

^on

^to

^and tl . may be estimated

similarly

with C

replaced by Ck .

^{To see this we} examine the

proof (see

^{for ex.}

[3]

or

[4])

^of the estimate

valid for

0 C to ~ t ~ t~,

^{and some} h &#x3E; 1. We notice that the same estimate holds for

complex

^t contained in ^a disk

and 6

sufficiently small, depending

^on

to,

ti ^{but not on}

t2.

^{Standard esti-}

mates for derivatives of

complex analytic

function then

give

^{the first esti-}

mate. The same

argument

^{can be}

applied

^to

DfG, yielding

^{different con-}

stants.

We now

proceed

^{as before.}

Using Stirling’s

^{formula we see that}

f(x, t)

⁼

L(q(t)G)

^{satisfies an esti-}

mate

for

0 &#x3E; 0,

^with

h(x) --&#x3E; 0

faster than any

negative

power of

lxl.

Using

the estimates for

t)

and its derivatives stated in the first

part

of section

4,

^{we see that}

fw(s, t)

and each s derivative

belongs

^to

y’(,g, t), uniformly in

^{co. The same} may be said for

where

~( ~ )

^{is the same} 000 cut-off function as in the case n = 1.

The

operator

defined

(as

in section

4) by

is

again parabolic, uniformly

^with

respect

to OJ, and varies

smoothy

^{with c~.}

We form

as in the case n ⁼ 1. The series involved will converge

absolutely

^and

uniformly

^on bounded s intervals and

uniformly

ⁱⁿ OJ, and the same holds

for the differentiated series. We use here the fact that because of the para-

bolicity assumption,

^the

operator A.(D,,)

^{has its}

highest

order coefficient bounded away from zero,

uniformly

in c~, from which the estimate for

a~,,L 1(s)

(analogous

^{to the one in section}

5)

^{can be made}

uniform in

^(o.

The desired fundamental solution is then

We have thus

proved

THEOREM 3. Let L

= A(Dx) - alat

^{be a}

parabolic operator (as defined

earlier in this

section)

^{in Rn X}

(-

^{00 t}

(0).

^{Then 0 there exists}

a

fundamental

solution with

support

^{in the}

strip

^{0 t} 8, which is Coo awacy

f rom

^the

origin.

The

following

theorem and

corollary

^were

proved by

Frank Jones for the heat

equation [7].

^{As he}

remarks,

^the

proof

remains valid for more

general parabolic equations having

fundamental solutions with

support

ⁱⁿ

0 t s.

Applying

^his

proof

^{to our}

operator

^{.L we obtain}

THEOREM 4. Let L ⁼

P(Dz) - alat

^be

parabolic

^{in the sense}

of

^{this sec-}

tion. Let

f belong

^to and suppose

supp f c Rn x [a, b).

^{Then there}

exists ^u in such that Lu ⁼

f

^{and supp u c Rn X}

[a, b].

COROLLARY. Let g be ^a continuous

f unction

^{on Rn and 8} &#x3E; 0. Then there exists a continuous

function u

^{on Ran X}

[0, (0)

^{such that}

BIBLIOGRAPHY

[1]

^L. EHRENPREIS, ^Fourier

analysis

^{in several}

complex

variables, ^New York, ^{N. Y.}

(1970).

[2]

H. O. FATTORINI - D. L. RUSSELL, ^Exact

controllability

^theorems

for

linear para- bolic

equations in

^one space dimension, Archive Rat. Mech. Anal., 4 (1971),

pp. 272-292.

[3]

^A. FRIEDMAN, Generalized

functions

^and

partial differential equations, Engle-

wood Cliffs, ^{N. J.} (1963).

[4]

I. M. GELFAND - G. E. SHILOV, Generalized Functions, vol. 3, ^Moscow

(1958).

[5] ^E. HOLMGREN, ^Sur

l’équation

^{de la}

propagation

^{de la chaleur,} Ark. Mat. Astr.

Physik,

⁴ (1908), pp. ^1-4.

[6] ^L.

HÖRMANDER,

^Linear

partial differential operators,

Academic Press Inc., New

York,

^{N. Y.}

(1963).

[7] B. F. JONES Jr., ^A

fundamental

^solution

for

^{the heat}

equation

^{which is}

supported

in a

strip,

^{J. Math.}

Analysis

^and

Applications,

⁶⁰

(1977),

pp. 314-324.

[8] ^D. LUDWIG, ^{The Radon}

transform

^on Euclidean spaces, Communications in Pure and

Applied

Math., ¹⁹

(1966),

pp. ^49-81.

[9]

^{D. L. RUSSELL, A}

unified boundary controllability theory for hyperbolic

^and

parabolic equations,

Studies in

Applied

Math., ⁵² (1973), pp. 189-211.

[10]

^T.

SEIDMAN, Boundary

observation and control

for

^{the heat}

equation:

^Dif-

ferential games and control

theory,

^Marcel Dekker, ^New York, ^{N. Y.} (1974),

pp. 321-351.